What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?
$\|x\| = |x|$ denotes the absolute value norm, which is a special case of the $L_1$ norm defined on the 1-D vector spaces formed by real or complex numbers.

$\|\textbf{x}\|_1 = \sum_{i=1}^n|x_i|$ denotes the Taxicab / Manhattan norm, relating to how a Taxi would drive along a rectangular grid of roads to reach a point $(x, y)$ from $(0,0)$.

$\|\textbf{x}\|_2 = \sqrt{x_1^2 + \dots + x_n^2}$ denotes the Euclidean norm on an N-D Euclidean space, which is a result of the Pythagorean theorem (the shortest distance between two points).